There are 100 passengers which are trying to get on a plane. The first passenger however has lost his boarding pass, so decides to sit on an arbitrary seat. Each successive passenger either sits on his own seat if it is empty, or on an arbitrary other if it is taken. What is the chance that the last person will sit on his own seat?

The chance is 50%. Indeed, the last passenger will either have to sit on his own seat, or the one which belongs to the first passenger. Since there hasn't be a preference made by anyone at any time towards any of these two seats, the probability that either of them is left last is 1/2.